Desicion Making in Multi-Objective Optimization for Industrial Applications - Data Mining and Visualization of Pareto Data

Optimization of engineering structures where multiple (more or less conflicting) objectives are simulta- neously considered, is getting more and more attractive in automotive industries. They usually involve a large number of design variables and the objectives are subject to certain constraints. Unlike single- objective problems, there are many trade-off solutions. The most common approach of using a single aggregate objective function (AOF), though simple, is not appropriate in most cases because a) it requires a priori information e.g., weights associated with each objective for weighted linear sum of the objectives method, that might not be available; and b) this approach yields a single trade-off solution instead of all possible trade-off solutions. Multi-objective evolutionary algorithms (MOEA) seem to be the best choice at the moment to overcome these issues. A set of solutions (Pareto data) is obtained as result, which reflect distinct trade-off so- lutions. A (optimal) decision needs to be taken to choose the most suitable trade-off among multiple conflicting objectives. Data mining and visualization techniques for high dimensional data provide helpful information to sub- stantially augment the decision making (alternative design selection) in multi-objective optimization en- vironment. A graphical approach to visualize the Pareto frontier is an intuitive and suitable approach to investigate the trade-off for three or fewer dimensions (objectives). However, it is not trivial to study relations in higher dimensions hence many visualization methods are proposed. The basic idea of these techniques is to reduce the dimensionality without loosing the relevant information required to recognize and understand relations and characteristics of the high dimensional Pareto data. Among the several developments in these fields, the Parallel Coordinates Plot (PCP), the Hyper-Radial Visualization (HRV), and the Self Organizing Maps (SOM) have been found the most promising. The parallel coordinates plot assigns one axis to each dimension and many dimensions are aligned in parallel. A data point is represented as a line connecting different axes. The HRV is based on a ra- dial calculation and transfers the multi-dimensional data to a two-dimensional data set by grouping the weighted objectives, that leads to a final solution with respect to the selected weights and the grouping. The designer incorporates his preferences by modifying the selection. The SOM algorithm projects the multi-dimensional Pareto data onto a two-dimensional map, whereby similar data is mapped to neigh- boring locations on the map. The lattices are color-coded to show the variation of the data on the map. The concepts of PCP, HRV and SOM are explained along with the various forms of visualization of Pareto data. All three approaches are investigated and respective pros and cons are identified using a shape optimization case crash application executed with LS-OPT. An implementation in the data mining and visualization framework D-SPEX is also provided.

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